Embedded newform 335.2.w.a.2.13

• Label: 335.2.w.a.2.13
• Level: $335$
• Weight: $2$
• Character: 335.2
• Analytic conductor: $2.675$
• Analytic rank: $0$
• Dimension: $1280$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 335 = 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	335.w (of order \(132\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.67498846771\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(32\) over \(\Q(\zeta_{132})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{132}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	335.2
Dual form			335.2.w.a.168.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.471157 - 0.518290i) q^{2} +(-0.637076 - 1.16672i) q^{3} +(0.143476 - 1.50255i) q^{4} +(-0.111677 + 2.23328i) q^{5} +(-0.304535 + 0.879897i) q^{6} +(-1.41662 + 4.43156i) q^{7} +(-1.96782 + 1.47309i) q^{8} +(0.666559 - 1.03719i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 38 q^{2} - 44 q^{3} - 44 q^{5} - 80 q^{6} - 38 q^{7} - 110 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/335\mathbb{Z}\right)^\times\).

\(n\)	\(136\)	\(202\)
\(\chi(n)\)	\(e\left(\frac{1}{66}\right)\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.471157	−	0.518290i	−0.333158	−	0.366487i	0.550324	−	0.834951i	\(-0.314504\pi\)
							−0.883482	+	0.468465i	\(0.844807\pi\)
\(3\)	−0.637076	−	1.16672i	−0.367816	−	0.673604i	0.626576	−	0.779360i	\(-0.284456\pi\)
							−0.994392	+	0.105756i	\(0.966274\pi\)
\(5\)	−0.111677	+	2.23328i	−0.0499436	+	0.998752i
							\(7\)	−1.41662	+	4.43156i	−0.535431	+	1.67497i	0.186050	+	0.982540i	\(0.440431\pi\)
−0.721481	+	0.692434i	\(0.756538\pi\)
\(11\)	−4.26056	+	1.47460i	−1.28461	+	0.444607i	−0.882209	−	0.470857i	\(-0.843945\pi\)
							−0.402399	+	0.915464i	\(0.631824\pi\)
\(13\)	−4.19533	−	0.501613i	−1.16358	−	0.139122i	−0.485582	−	0.874191i	\(-0.661392\pi\)
							−0.677993	+	0.735069i	\(0.737150\pi\)
\(17\)	0.747989	+	0.617592i	0.181414	+	0.149788i	0.722977	−	0.690872i	\(-0.242773\pi\)
							−0.541563	+	0.840660i	\(0.682167\pi\)
\(19\)	−1.65392	+	3.20815i	−0.379435	+	0.736000i	−0.998793	−	0.0491076i	\(-0.984362\pi\)
							0.619359	+	0.785108i	\(0.287393\pi\)
\(23\)	7.61377	+	4.64083i	1.58758	+	0.967681i	0.985585	+	0.169181i	\(0.0541122\pi\)
							0.601995	+	0.798500i	\(0.294373\pi\)
\(29\)	−1.16157	−	0.670634i	−0.215698	−	0.124534i	0.388259	−	0.921551i	\(-0.373077\pi\)
							−0.603957	+	0.797017i	\(0.706410\pi\)
\(31\)	3.78814	−	4.81701i	0.680369	−	0.865160i	−0.316134	−	0.948715i	\(-0.602385\pi\)
							0.996503	+	0.0835548i	\(0.0266274\pi\)
\(37\)	1.25681	−	4.69047i	0.206618	−	0.771109i	−0.782332	−	0.622862i	\(-0.785970\pi\)
							0.988950	−	0.148248i	\(-0.0473633\pi\)
\(41\)	−0.317620	+	0.226176i	−0.0496039	+	0.0353227i	−0.604590	−	0.796537i	\(-0.706663\pi\)
							0.554986	+	0.831860i	\(0.312724\pi\)
\(43\)	−5.92607	+	2.21031i	−0.903717	+	0.337069i	−0.757998	−	0.652257i	\(-0.773822\pi\)
							−0.145719	+	0.989326i	\(0.546550\pi\)
\(47\)	−1.06987	−	0.0254676i	−0.156056	−	0.00371483i	−0.0543031	−	0.998525i	\(-0.517294\pi\)
							−0.101753	+	0.994810i	\(0.532445\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	335.2.w.a.2.13	✓	1280
5.3	odd	4	inner	335.2.w.a.203.13	yes	1280
67.34	odd	66	inner	335.2.w.a.302.13	yes	1280
335.168	even	132	inner	335.2.w.a.168.13	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
335.2.w.a.2.13	✓	1280	1.1	even	1	trivial
335.2.w.a.168.13	yes	1280	335.168	even	132	inner
335.2.w.a.203.13	yes	1280	5.3	odd	4	inner
335.2.w.a.302.13	yes	1280	67.34	odd	66	inner